SLIDE 1 A brief introduction to quasiparticles in frustrated magnets
Claudio Castelnovo
TCM group Cavendish Laboratory University of Cambridge
17-06-2020 CMP in All the Cities
SLIDE 2 Outline
◮ magnetism, frustration and spin liquid behaviour ◮ modelling spin liquids: general overview ◮ quasiparticle excitations: 6-vertex and 8-vertex model
- classical behaviour: deconfinement, fractionalisation,
dynamical constraints and entropic interactions
- quantum behaviour: fractional statistics and dual
quasiparticles → toric code and quantum spin ice
◮ quantum spin liquids at finite temperature (a prelude to the second talk) ◮ conclusions
SLIDE 3 Conventional Magnetism vs Spin Liquids
T / J ~ 1
“trivial” disorder
“trivial”: T J ⇒ high-T expansion holds (SiSj ∼ −Hij/T) frustration: inability to minimise locally all energy terms ⇒ Tc ≪ J H = J
ij SiSj
(triang. Ising AFM)
SLIDE 4 Conventional Magnetism vs Spin Liquids
T / J ~ 1
“trivial” disorder
“trivial”: T J ⇒ high-T expansion holds (SiSj ∼ −Hij/T) frustration: inability to minimise locally all energy terms ⇒ Tc ≪ J
T / J ~ 1
disorder
?
T /J << 1 c
SLIDE 5 Conventional Magnetism vs Spin Liquids
T / J ~ 1
disorder
?
T /J << 1 c
◮ ∼ 1 is typically a crossover (Schottky anomaly) ◮ no long range order ◮ non-trivial spin correlations (T < J ⇒ SiSj ∼ −Hij/T) − → spin liquid
SLIDE 6
Modelling (classical) spin liquids
example: nn Ising AFM on triangular lattice 2:1 triangles vs 3:0 triangles energy difference: ∆ ∼ J ⇒ projects onto mostly 2:1 configurations for T ∆
SLIDE 7 Modelling (classical) spin liquids
generally: H ∼ H∆ + Hδ ◮ leading contribution (H∆) projects onto subset of configuration space (no spontaneous symmetry breaking) for T ∆ ◮ possible subleading contributions (Hδ) cause ordering for T δ ≪ ∆ (triang. nn Ising AFM: Hδ = 0)
T / ~ 1
disorder
SL
/ << 1 D d D
SLIDE 8 Effective dimer description
for T ∆, mostly 2:1 triangles ferro bonds equivalent to dimers
leading to: ◮ extensive degeneracy ◮ non-trivial correlations
SLIDE 9 Emergent gauge symmetry and dipolar correlations
A B
dimer = flux 2 from A to B no-dimer = flux 1 from B to A dimer constraint = divergenceless condition
− → emergent gauge field
Henley AR 2010
⇒ 2D dipolar correlations: flux flux ∼ dimer dimer ∼ S S
SLIDE 10 Elementary excitations of H∆
for convenience: Ising model on bonds of square lattice
A B A B +
+ + + + + +
consider Hamiltonians that result in leading projection term H∆ favouring: ◮
i∈s σi = 0 (6 vertex model)
◮
i∈s σi = 1 (8 vertex model)
+ +
+
+
+ + +
+ s i s i
SLIDE 11 Six-vertex vs eight-vertex model
◮ extensively degenerate
Pauling’s entropy estimate: 2N spins, N sites n out of 16 (n = 6, 8) minimal energy configurations per site S ∼ ln
16
N ∼ sn N
◮ unusual correlations
6-vertex model: σi = ±1 ⇔ flux from A to B (B to A) ⇒ divergenceless condition and dipolar correlations
[Isakov PRL 2004]
8-vertex model: plaquette flips preserve minimal energy ⇒ zero-range corr. σiσj = 0, ∀i = j but topological properties
SLIDE 12 Excitations in the 8-vertex model
spin flip on ground state ⇓ two defects:
i∈s σi = −1
+
+ + + + + + + + + + + + + + +
- ◮ spins next to defect flip at no energy cost (hop or annihilate)
◮ trivially deconfined ◮ elementary excitations are single defective sites ◮ lattice gas of (RW) particles that pair create or annihilate
SLIDE 13 Excitations in the 8-vertex model
spin flip on ground state ⇓ two defects:
i∈s σi = −1
+
+ + + + + + + + + + + + + +
- ◮ spins next to defect flip at no energy cost (hop or annihilate)
◮ trivially deconfined ◮ elementary excitations are single defective sites ◮ lattice gas of (RW) particles that pair create or annihilate
SLIDE 14 Excitations in the 8-vertex model
spin flip on ground state ⇓ two defects:
i∈s σi = −1
+ + + + + + + + + + + + + + + +
◮ spins next to defect flip at no energy cost (hop or annihilate) ◮ trivially deconfined ◮ elementary excitations are single defective sites ◮ lattice gas of (RW) particles that pair create or annihilate
SLIDE 15 Excitations in the 8-vertex model
spin flip on ground state ⇓ two defects:
i∈s σi = −1
+ + + + + + + + + + + + + + +
- +
- ◮ spins next to defect flip at no energy cost (hop or annihilate)
◮ trivially deconfined ◮ elementary excitations are single defective sites ◮ lattice gas of (RW) particles that pair create or annihilate
SLIDE 16 Excitations in the 6-vertex model
spin flip on ground state ⇓ two defects:
i∈s σi = ±1
+
+ + + + + + + + + + +
◮ spins next to defect flip at no energy cost only along alternating sign paths (+ − + − + − . . .) ◮ elementary excitations are single defective sites (deconfined) ◮ constrained gas of particles that pair create or annihilate
SLIDE 17 Excitations in the 6-vertex model
spin flip on ground state ⇓ two defects:
i∈s σi = ±1
+
+ + + + + + + + + + + + +
◮ spins next to defect flip at no energy cost only along alternating sign paths (+ − + − + − . . .) ◮ elementary excitations are single defective sites (deconfined) ◮ constrained gas of particles that pair create or annihilate
SLIDE 18 Excitations in the 6-vertex model
spin flip on ground state ⇓ two defects:
i∈s σi = ±1
+ + + + + + + + + + + + + +
◮ spins next to defect flip at no energy cost only along alternating sign paths (+ − + − + − . . .) ◮ elementary excitations are single defective sites (deconfined) ◮ constrained gas of particles that pair create or annihilate
SLIDE 19 Excitations in the 6-vertex model
spin flip on ground state ⇓ two defects:
i∈s σi = ±1
+ + + + + + + + + + + + + +
◮ spins next to defect flip at no energy cost only along alternating sign paths (+ − + − + − . . .) ◮ elementary excitations are single defective sites (deconfined) ◮ constrained gas of particles that pair create or annihilate
SLIDE 20
Excitations in the 6-vertex model (gauge flux rep.)
spin flip on ground state ⇓ pair of oppositely charged defects: sinks (3i1o) and sources (3o1i) of gauge flux ◮ defects move freely along oriented arrow paths ◮ close interplay: spins determine how defects move; defect motion rearranges spins ◮ lattice gas of gauge charges → spins mediate Coulomb int.
SLIDE 21
Excitations in the 6-vertex model (gauge flux rep.)
spin flip on ground state ⇓ pair of oppositely charged defects: sinks (3i1o) and sources (3o1i) of gauge flux ◮ defects move freely along oriented arrow paths ◮ close interplay: spins determine how defects move; defect motion rearranges spins ◮ lattice gas of gauge charges → spins mediate Coulomb int.
SLIDE 22
Excitations in the 6-vertex model (gauge flux rep.)
spin flip on ground state ⇓ pair of oppositely charged defects: sinks (3i1o) and sources (3o1i) of gauge flux ◮ defects move freely along oriented arrow paths ◮ close interplay: spins determine how defects move; defect motion rearranges spins ◮ lattice gas of gauge charges → spins mediate Coulomb int.
SLIDE 23
Excitations in the 6-vertex model (gauge flux rep.)
spin flip on ground state ⇓ pair of oppositely charged defects: sinks (3i1o) and sources (3o1i) of gauge flux ◮ defects move freely along oriented arrow paths ◮ close interplay: spins determine how defects move; defect motion rearranges spins ◮ lattice gas of gauge charges → spins mediate Coulomb int.
SLIDE 24 Parenthesis: entropic Coulomb interaction
◮ no energetic interactions between defects ◮ yet probability P(R) of two oppositely charged defects R apart ∼ exp[Cd(R)] with Coulomb potential Cd(R) in d dim. ◮ ⇒ entropic Coulomb interaction −T Cd(R)
0.1 0.2 0.3 0.4 10
−2.54
10
−2.51
10
−2.48
10
−2.45
inverse monopole separation (units of pyrochlore a) distribution function
d = 3, Cd(R) ∼ 1/R
CC et al. PRB 2011
SLIDE 25 Low temperature (classical) dynamics
behaviour controlled by sparse defect motion: ◮ 8-vertex: 2D random walk + pair creation/annihilation events (aka reaction-diffusion process) ◮ 6-vertex: constrained lattice gas motion + entropic Coulomb interactions
Toussaint et al. J. Chem. Phys. 1983 Ginzburg et al. PRE 1997 Ryzhkin et al. JETP 2005, EPL 2013 CC et al. PRL 2010, PRB 2019
SLIDE 26
Quantum spin liquids
H = H∆ + Hδ where Hδ ∼ Hdefect int. +Hdefect hopping ◮ neglect Hdefect int. for simplicity ◮ hopping t ∆ → defect dynamics (first order) + ‘ground state’ dynamics (perturbatively: ∆ ∼ t (t/∆)n)
ring exchange
D
SLIDE 27 Quantum spin liquids
t
disordered paramagnet
energy
?
D D
temp.
classical SL quantum SL
<< t
energy D 2D t ~ ~ t
two-defect sector zero-defect sector
SLIDE 28 Intermediate regime (∆ T < t)
◮ intermediate between classical and quantum behaviour ◮ highest temperature where precursor QSL behaviour may appear (→ experiments) ◮ underlying spins act as self-generated disorder in defect motion → localisation ◮ general framework hitherto unavailable... (but interesting case studies)
arXiv:1909.08633 arXiv:1911.06331 arXiv:1911.05742 arXiv:2005.03036
→ next talk (Thu 25th June, 16:30)
SLIDE 29 Quantum 8-vertex model
(aka toric code in a field)
+ +
+
i p H∆ = −∆
σz
i
Ht = −t
σx
i
⇒ H∆ = −∆
σx
i ,
∆ ∼ t4 ∆3 H∆ + H∆ = toric code
Kitaev 2003
◮ H∆ favours
i∈s σz i = +1
◮ H∆ favours
i∈p σx i = +1
◮ they commute and can be simultaneously satisfied
SLIDE 30 Quantum 8-vertex model
(aka toric code in a field)
elementary excitations: ◮ star defects (
i∈s σz i = −1, cost ∼ ∆)
◮ plaquette defects (
i∈p σx i = −1, cost ∼ ∆)
point-like, deconfined bosons, with mutual semionic statistics
s p
= -
s p
t D D
temp.
{
◮ star defects: sparse and hop coherently ◮ plaquette defects: thermally populated (dense and incoherent)
SLIDE 31 Quantum 8-vertex model (∆ T < t)
incoherent superposition of plaquette defects (ensemble average) + coherent star defect hopping
p p p p p p p
tight-binding charges in a random π-flux background: Anderson localisation of emergent particles (+ thermodynamic response due to mutual statistics)
SLIDE 32 Quantum 6-vertex model
(aka quantum spin ice in a field)
+ +
+
i p H∆ = −∆
σz
i
2 Ht = −t
σx
i
⇒ H∆ = −∆
1 σ− 2 σ+ 3 σ− 4 + h.c.
(∆ ∼ t4/∆3) H∆ + H∆ = quantum (square) spin ice
Hermele et al. 2004
◮ H∆ favours
i∈s σz i = 0
◮ H∆ favours + − +− (‘flippable’) plaquettes ◮ they do not commute and cannot be simultaneously satisfied
SLIDE 33 Quantum 6-vertex model
(aka quantum spin ice in a field)
elementary excitations: ◮ star defects (
i∈s σz i = ±1, cost ∼ ∆, gauge charge ±1)
◮ plaquette dynamics promotes gauge symm. to QED ◮ plaq. defects: dual charges (cost ∆, not trivially related to H∆) ◮ gapless photons
Hermele et al. 2004
point-like, deconfined quasiparticles, with electromag. interactions
(and not immediately obvious statistics) t D D
temp.
{
◮ star defects: sparse and hop coherently ◮ dual charges and photons: thermally populated
SLIDE 34
Quantum 6-vertex model (∆ T < t)
working assumption: incoherent superposition of underlying spins (ensemble average) + coherent star defect hopping constrained dynamics ↔ tight-binding on a random network ↔ (emergent) configurational disorder
SLIDE 35 Conclusions
◮ frustration in magnetic systems opens a window into unusual and interesting spin liquid phases ◮ powerful effective modelling in terms of interplay between spin (GS) vacuum and quasiparticle excitations
- classical: emergent symmetries, fractionalisation,
reaction-diffusion processes and entropic interactions
- quantum: dual quasiparticles and non-trivial statistics
◮ tease: interesting intermediate temperature regime with potential precursor signatures of QSL behaviour at lower temperatures
